# Let W be the subspace of all diagonal matrices in M_{2,2}. Find a bais for W. Then give the dimension of W. If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix

Question
Matrices
Let W be the subspace of all diagonal matrices in $$M_{2,2}$$. Find a bais for W. Then give the dimension of W.
If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix

2021-01-29
Step 1
Given that W is the subspace of all diagonal matrices in $$M_{2,2}$$
The objective is to a basis for W.
Step 2
Consider the given vector space W of all diagonal matrices in $$M_{2,2}$$
Therefore,
$$W=\left\{\begin{bmatrix}a & 0 \\0 & b \end{bmatrix}, \text{a and b can be any real number}\right\}$$
Let E be basis for W.
Therefore,
$$\begin{bmatrix}a & 0 \\0 & b \end{bmatrix}=a\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix}+b\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}$$
Now, let $$\lambda_1$$ and $$\lambda_2$$ be any two scalars such that:
$$\lambda_1\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix}+\lambda_2\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}=[]$$
$$\begin{bmatrix}\lambda_1 & 0 \\0 & 0 \end{bmatrix}+\begin{bmatrix}0 & 0 \\0 & \lambda_2 \end{bmatrix}=\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix}$$
$$\begin{bmatrix}\lambda_1 & 0 \\0 & \lambda_2 \end{bmatrix}=\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix}$$
Equating the elements:
$$\lambda_1=0 ,\lambda_2=0$$
Therefore, $$\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix}$$ and $$\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}$$ are linear independent.
Hence, the basis of W is $$E=\left\{\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix},\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}\right\}$$ and dimension is 2.

### Relevant Questions

Let B be a $$4 \times 4$$ matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (so that the column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product ABC (same B) of three matrices.
Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.
The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P $$(A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).$$
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
Enter the expression that would produce the answer (do include the answer) for row 1 column 1 of the multiplied matrix $$A \cdot B$$:
List the expression in order with the original values using $$\cdot$$ for multiplication.
then find $$A \cdot B$$
If $$A=\begin{bmatrix}3 & 7 \\2 & 4 \end{bmatrix} \text{ and } B=\begin{bmatrix}-3 & 6 \\4 & -2 \end{bmatrix}$$
Let $$M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})$$ be the set of 2 x 2 matrices with the entries in $$\mathbb{Z}/\mathbb{6Z}$$
a) Can you find a matrix $$M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})$$ whose determinant is non-zero and yet is not invertible?
b) Does the set of invertible matrices in $$M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})$$ form a group?
In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
Determine whether the subset of $$M_{n,n}$$ is a subspace of $$M_{n,n}$$ with the standard operations. Justify your answer.
The set of all $$n \times n$$ matrices whose entries sum to zero
Give a full correct answer for given question 1- Let W be the set of all polynomials $$\displaystyle{a}+{b}{t}+{c}{t}^{{2}}\in{P}_{{{2}}}$$ such that $$\displaystyle{a}+{b}+{c}={0}$$ Show that W is a subspace of $$\displaystyle{P}_{{{2}}},$$ find a basis for W, and then find dim(W) 2 - Find two different bases of $$\displaystyle{R}^{{{2}}}$$ so that the coordinates of $$\displaystyle{b}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{5}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$ are both (2,1) in the coordinate system defined by these two bases
Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for $$U^\perp$$ where $$U^{\perp}\left\{A \in V |(A|B)=0 \forall B \in U \right\}$$